I'm testing a 4-factor solution against a second-order CFA with 4 lower-order factors. My sample sizes range from 1,000-9,000+, so I'd imagine that the chi-square diff tests are overpowered and much too sensitive to trivial differences.

My questions: - Are there any large-sample alternatives to the chi-square difference test? - Or can I simply adopt a more stringent criterion (maybe p<.001 or p<.0001)? - Finally, can you point me to any articles on this topic?